Question: Given a cylinder of fixed volume $V,$ the total surface area (including the two circular ends) is minimized for a radius of $R$ and height $H.$  Find $\frac{H}{R}.$
Explanation: For radius $r$ and height $h,$ the volume is given by $\pi r^2 h = V,$ and the total surface area is given by
\[A = 2 \pi r^2 + 2 \pi rh.\]By AM-GM,
\begin{align*}
A &= 2 \pi r^2 + 2 \pi rh \\
&= 2 \pi r^2 + \pi rh + \pi rh \\
&\ge 3 \sqrt[3]{(2 \pi r^2)(\pi rh)(\pi rh)} \\
&= 3 \sqrt[3]{2 \pi^3 r^4 h^2}.
\end{align*}Since $\pi r^2 h = V,$ $r^2 h = \frac{V}{\pi}.$  Then
\[3 \sqrt[3]{2 \pi^3 r^4 h^2} = 3 \sqrt[3]{2 \pi^3 \cdot \frac{V^2}{\pi^2}} = 3 \sqrt[3]{2 \pi V^2}.\]Equality occurs when $2 \pi r^2 = \pi rh,$ so $\frac{h}{r} = \boxed{2}.$